Brady, 34, had been in jail on $500,000 bail since January, when she was charged with assaulting Rehma Sabir in the child’s ­Cambridge home. Rehma died two days later, on Jan. 16.

Last week, a grand jury indicted Brady on murder charges after the medical examiner’s office ruled the child’s death a homicide.

On Thursday, prosecutors reiterated their case that the child was in Brady’s exclusive care at the time she suffered the blunt force injuries that caused her death. Patrick Fitzgerald, an assistant Middlesex district attorney, said the injuries were consistent with “violent shaking and direct impact to the head.”

A specialist who examined the child said there was “no other expla­nation” for her injuries, Fitzgerald told the judge.

According to Fitzgerald, Brady told investigators that the child slept much of the day of the alleged assault. When she was awakened for lunch, she appeared lethargic and only drank half her milk, Brady told them.

That afternoon, Brady and the child’s grandmother found the child limp and unresponsive in her crib. At the hospital, she was found to have suffered massive bleeding in her brain and the back of her eyes.

Rehma’s parents have request­ed privacy and did not appear in court.

Arguing that Brady should be held without bail, Fitzgerald described Brady, a native of ­Ireland who has been living in the country illegally, as a flight risk. If she were to make bail and be deported, the government of Ireland would not ­return her to the United States because the charge against her is a capital crime, he said.

“If she is returned to Ireland, we will not be able to extradite her,” Fitzgerald said.

But Brady’s lawyer, Melinda Thompson, said her client would agree not to leave the country if she were released.

“She wants to answer to these charges,’’ Thompson said. “She wants to clear her name.”

After the child was hospitalized, Brady kept vigil with the family and fully cooperated with investigators. For days ­before her arrest, she stayed at home, Thompson said.

“She didn’t leave for all that time,” she said.

As she has at previous court hearings, Thompson forcefully declared that her client did not commit the crimes of which she is accused.

“My client maintains her inno­cence,” she said. “The wrong person has been charged with this crime. She did not hurt that child.”

Brady had been baby-sitting for Rehma since last July. She also looked after a second child part time, Fitzgerald said.

Thompson pointed out that Brady did not care for the child from Dec. 7 to Jan. 2, when the family was traveling overseas, and that tests had shown the child had numerous preexisting injuries, fractures that were between two weeks and two months old.

The timeline suggests that others may have harmed the child, the lawyer said. The child also had been ill and was diagnosed with malnutrition before her death, according to Thompson.